| L(s) = 1 | + 3.18·2-s + 20.6·3-s − 117.·4-s − 477.·5-s + 65.7·6-s + 883.·7-s − 782.·8-s − 1.76e3·9-s − 1.52e3·10-s − 5.16e3·11-s − 2.43e3·12-s + 5.64e3·13-s + 2.81e3·14-s − 9.86e3·15-s + 1.25e4·16-s − 5.86e3·17-s − 5.60e3·18-s + 6.85e3·19-s + 5.62e4·20-s + 1.82e4·21-s − 1.64e4·22-s − 3.71e4·23-s − 1.61e4·24-s + 1.50e5·25-s + 1.79e4·26-s − 8.15e4·27-s − 1.04e5·28-s + ⋯ |
| L(s) = 1 | + 0.281·2-s + 0.441·3-s − 0.920·4-s − 1.70·5-s + 0.124·6-s + 0.973·7-s − 0.540·8-s − 0.804·9-s − 0.480·10-s − 1.16·11-s − 0.406·12-s + 0.712·13-s + 0.274·14-s − 0.754·15-s + 0.768·16-s − 0.289·17-s − 0.226·18-s + 0.229·19-s + 1.57·20-s + 0.430·21-s − 0.328·22-s − 0.636·23-s − 0.238·24-s + 1.92·25-s + 0.200·26-s − 0.797·27-s − 0.896·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - 6.85e3T \) |
| good | 2 | \( 1 - 3.18T + 128T^{2} \) |
| 3 | \( 1 - 20.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 477.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 883.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.64e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.86e3T + 4.10e8T^{2} \) |
| 23 | \( 1 + 3.71e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.87e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.42e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.67e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.57e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.39e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.80e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.59e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.48e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.06e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.04e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.10e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.75e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.64e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.92278800207290608557332042392, −14.85889372754534975778890571104, −13.76327572637728651946830856810, −12.20255731670735797416615345243, −10.95082667044089614081906507840, −8.572233735888957194970151729705, −7.947937172009549844828116499513, −4.99695903503744192178456194094, −3.50683639537017352707305423611, 0,
3.50683639537017352707305423611, 4.99695903503744192178456194094, 7.947937172009549844828116499513, 8.572233735888957194970151729705, 10.95082667044089614081906507840, 12.20255731670735797416615345243, 13.76327572637728651946830856810, 14.85889372754534975778890571104, 15.92278800207290608557332042392
